This is achieved by expressing \(\mathrm {Var}(\mathcal {N}_{K,x})\)  in terms of zeroes of the family of Hecke \(L\) -functions corresponding to the characters given by \(\Xi _k(\alpha )=(\alpha /\overline{\alpha })^{2k}\) , \(k \in \mathbb {Z}\)  for an ideal \((\alpha )\)  in \(\mathbb {Z}[i]\) . The result then follows from an upper bound for said variance. The distribution of values of this family of Hecke \(L\) -functions was studied by Duke, Friedlander, and Iwaniec in [1]}. Their results suggest that this family can be modeled with a symplectic regime. Indeed, Rudnick and Waxman develop a random matrix model that they use to conjecture the following.
